1. Destruction of adiabatic invariance at resonances in slow-fast Hamiltonian systems Аnatoly Neishtadt Space Research Institute, Moscow aneishta@iki.rssi.ru

2. А diabatic invariant is an approximate first integral of the system with slow and fast variables (slow-fast system). e B d l

3. If a system has enough number of adiabatic invariants then the motion over long time intervals is close to a regular one. Destruction of adiabatic invariance is one of mechanisms of creation of chaotic dynamics.

4. System with rotating phases: (slow) (fast) averaging I(x) is a first integral of the averaged system => it is an adiabatic invariant of the original system

5. x - resonant surface -trajectory of the averaged system are integer numbers

6. Slow-fast Hamiltonian system: slow variables fast phases

7. averaging (adiabatic approximation)

8. I p q resonant surface I = const adiabatic trajectory capture escapescattering

9. Two-frequency systems: Effect of each resonance can be studied separately.

10. А. Partial averaging for given resonance. Canonical transformation: Averaging over Hamiltonian: is the resonant phase

11. B. Expansion of the Hamiltonian near the resonant surface. R q p

12. - resonant flow

13. Dynamics of (resonant phase) and (deviation from the resonant surface) is described by the pendulum-like Hamiltonian: pendulum with a torque and slowly varying parameters

14. Phase portraits of pendulum-like system P P

15. Capture: Probability of capture:

16. I n-out function: “inner adiabatic invariant” = const

17. Scattering on resonance. Value should be treated as a random variable uniformly distributed on the interval

18. Results of consequent passages through resonances should be treated as statistically independent according to phase expansion criterion.

19. Resonance: Example: motion of relativistic charged particle in stationary uniform magnetic field and high-frequency harmonic electrostatic wave (A.Chernikov, G.Schmidt, N., PRL, 1992; A.Itin, A.Vasiliev, N., Phys.D, 2000). Larmor circle wave Capture into resonance means capture into regime of surfatron acceleration (T.Katsouleas, J.M.Dawson, 1985)

20. B k Assumptions:

21. After rescaling: After transformation: Conjugated variables:

22. I q p Resonant surface:Resonant flow:

23. Hamiltonian of the “pendulum”:

24. Trajectory of the resonant flow is an ellipse.

25. Capture into resonance and escape from resonance:

26. Trajectory of the resonant flow is a hyperbola. Condition of acceleration:

27. Capture into resonance (regime of unlimited surfatron acceleration):

28. Scattering on resonance: